The challenge now is to solve Eqs. () subject to Eqs. (REF ) and (REF ). For the blast problem, the condition of conservation of energy allows us to determine exactly the exponent \(\delta \)  and the constant \(\alpha \) , and also a closed form solution for the scaling fields [1]}, [2]}, [3]} — this is an example of a self-similar solution of the first kind [4]}. In the “one-sided” splash problem, energy is not conserved in the full hydrodynamic regime as it is lost to the splatter particles [5]}, [6]}. In this case it turns out that one has to treat the system Eqs. (,REF ,REF ) as a non-linear eigenvalue equation. Only for a unique choice of \(\Delta \)  (and therefore \(\delta \) ) are the boundary conditions satisfied — this type of solutions are known in the literature as self-similar solutions of the second kind [4]}, first discovered by Guderley [8]}.
